Measurements are never perfectly exact, so any quantity calculated from measurements also has uncertainty. Calculus gives a fast way to estimate how small input errors affect an output. The key idea is sensitivity, which tells how strongly a function responds to a small change in its input.
This matters in physics, engineering, chemistry, and data analysis whenever measured values are used in formulas.
For a computed quantity y = f(x), the differential dy = f'(x) dx approximates the output error caused by a small input error dx. If the function changes steeply near the measured value, a small measurement error can create a large output error. Relative error compares the error size to the value itself, and percentage error expresses that comparison as a percent.
For functions of several variables, partial derivatives show how uncertainty from each input contributes to the final uncertainty.
Key Facts
- For one variable, dy = f'(x) dx estimates the change in y caused by a small change dx.
- Absolute error in y is often estimated by |dy| = |f'(x)| |dx|.
- Relative error is approximately |dy| / |y|, where y = f(x).
- Percentage error is relative error times 100%, so percentage error = (|dy| / |y|) x 100%.
- For y = f(x1, x2, ..., xn), the maximum estimated absolute error is |dy| <= |fx1| |dx1| + |fx2| |dx2| + ... + |fxn| |dxn|.
- For products and powers, relative errors often combine simply, such as if y = x^n then |dy| / |y| ≈ |n| |dx| / |x|.
Vocabulary
- Differential
- A differential is a small change estimate, such as dy = f'(x) dx, used to approximate how a function changes.
- Absolute error
- Absolute error is the estimated size of the uncertainty in a value, measured in the same units as the value.
- Relative error
- Relative error is the absolute error divided by the magnitude of the value being measured or computed.
- Percentage error
- Percentage error is the relative error multiplied by 100%.
- Sensitivity
- Sensitivity describes how much the output of a function changes in response to a small change in an input.
Common Mistakes to Avoid
- Using dy as the exact error, which is wrong because dy is a linear approximation that is best for small errors near the measured value.
- Forgetting the absolute value in error estimates, which is wrong because error size should be nonnegative even if the derivative is negative.
- Confusing absolute error with relative error, which is wrong because absolute error has units while relative error is a unitless fraction.
- Applying percentage error before computing the output value, which is wrong because percentage error in y must compare the estimated output error to |y|.
Practice Questions
- 1 A cube has side length x = 5.00 cm with possible error dx = 0.02 cm. Using V = x^3 and differentials, estimate the absolute error and percentage error in the volume.
- 2 The period of a pendulum is modeled by T = 2π sqrt(L / g). If g is treated as exact, L = 0.800 m, and dL = 0.004 m, estimate the relative error and percentage error in T.
- 3 A function has a very small derivative near the measured input value. Explain how that affects the propagated error in the computed output, and give a physical or mathematical example.